How logical foundation for quantum theory derives from foundational anomalies in pure mathematics

Abstract.  Logical foundation for quantum theory is considered. I claim that quantum theory correctly represents Nature when mathematical physics embraces and indeed features, logical anomalies inherent in pure mathematics.

This approach reveals a connection linking undecidability in arithmetic with the logic of quantum experiments. The undecidablity occupies an algebraic environment which is the missing foundation for the 3-valued logic predicted by Hans Reichenbach, shown by him to resolve ‘causal anomalies’ of quantum mechanics, such as: inconsistency between prepared and measured states, complementarity between pairs of observables, and the EPR-paradox of action at a distance.

Arithmetic basic to mathematical physics, is presented formally as a logical system consisting of axioms and propositions. Of special interest are all propositions asserting the existence of particular numbers. All numbers satisfying the axioms permeate the arithmetic indistinguishably, but these logically partition into two distinct sets: numbers whose existence the axioms determine by proof, and numbers whose existence axioms cannot determine, being neither provable nor negatable.

Failure of mathematical physics to incorporate this logical distinction is seen as reason for quantum theory being logically at odds with quantum experiments. Nature is interpreted as having rules isomorphic to the abovementioned axioms with these governing arithmetical combinations of necessary and possible values or effects in experiments. Soundness and Completeness theorems from mathematical logic emerge as profoundly fundamental principles for quantum theory, making good intuitive sense of the subject.

Full Article: http://www.vixra.org/pdf/1101.0045v6.pdf

Indeterminacy in arithmetic, well-known to logicians but missing from quantum theory

Abstract. This article is one of a series explaining the nature of mathematical undecidability discovered within quantum theory. Crucially, a formula’s undecidability certifies its indeterminacy and vice versa. This paper describes the algebraic environment in which the undecidability and indeterminacy originate; provides proof of their existence; and demonstrates the role these play in a three-valued logic which is free to permeate mathematical physics via this algebra.
The radical idea applied in this research is taken from very well-known results in mathematical logic. All scalars engage in the arithmetic of scalars by way of a single algebra. But in terms of validity, these scalars partition into sets which are logically distinct: those with valid existence with respect to this algebra, and those with indeterminate existence. Failure of mathematical physics to notice this distinction is the reason why quantum theory is logically at odds with quantum experiments.

http://www.vixra.org/pdf/1101.0045v3.pdf

Quantum indeterminacy is found sensitive to scaling and seen to vanish at the macroscopic

PrePrint posted at: http://www.vixra.org/abs/1101.0097

Abstract:

Standard methods of quantum theory are employed, excepting: quantum theory is initialised by the a priori adoption of the Field Axioms; and the square root of minus one is not introduced initially as if axiomatic. Its adoption is postponed until inconsistency in the theory forces its introduction. Entry of this scalar, logically independent of the Axioms, relieves the inconsistency but introduces mathematical undecidability and indeterminacy. Nevertheless, indeterminate formulae derive determinate probability along with Pythagorean addition. Orthogonality is indicated as the condition around which logical anomalies in quantum physics hinge.

pdf: http://www.vixra.org/pdf/1101.0097v1.pdf

 

Gödelian Features Found at Quantum Indeterminacy: Inconsistency, Undecidability and Self Reference

PrePrint posted at: http://www.vixra.org/abs/1101.0075

Abstract:

Standard methods of quantum theory are employed, excepting: quantum theory is initialised by the a priori adoption of the Field Axioms; and the square root of minus one is not introduced initially as if axiomatic. Its adoption is postponed until inconsistency in the theory forces its introduction. Entry of this scalar, logically independent of the Axioms, relieves the inconsistency but introduces mathematical undecidability and indeterminacy. Nevertheless, indeterminate formulae derive determinate probability along with Pythagorean addition. Orthogonality is indicated as the condition around which logical anomalies in quantum physics hinge.

pdf: http://www.vixra.org/pdf/1101.0075v1.pdf

Indeterminate scalars under the Field Axioms: foundation for Reichenbach’s quantum logic

Paper newly posted at: http://www.vixra.org/abs/1101.0045

Abrtact:

Inherent within quantum measurement experiments is a decision process which current theory fails to express and does not explain. Each act of measurement indeterminately decides on one outcome from a spectrum of values. Nature executes this decision, but evidence indicates that the elected outcome is not caused by any physical influence.

The discrepancy between experiment and theory is traceable to a logical detail of arithmetic, not encoded in mathematical physics. Mathematical physics deals in semantic theories. These disregard this logical detail. For classical physics, no discrepancy is evident; in that domain, semantic truth and logical validity coincide. Happily, a logical implementation of the arithmetic, rather than a semantic one, does encode the indeterminate details .

The arithmetic in question is that of scalars. These are mathematical objects whose rules of algebra are the Field Axioms. Mathematical physics assumes the a priori existence of scalars. But in this article, apriority is transferred to the Field Axioms themselves. This step elevates the semantic theory to a logical one. Model Theory, a branch of mathematical logic, sets the Field Axioms within a rigorous environment that naturally differentiates between scalars they derive, distinct from scalars that satisfy them. A theorem is proved which isolates scalars whose existence is neither provable nor disprovable. These are the scalars which satisfy Axioms, but are not derivable; they are mathematically undecidable and logically indeterminate. Examples are worked through. The scalars’ modes of existence furnish the 3-valued logic which is foundation for Hans Reichenbach’s quantum logic.

pdf: http://www.vixra.org/pdf/1101.0045v1.pdf

The logical indeterminacy and mathematical undecidability under the Field Axioms

Foundation of Reichenbach’s quantum logic

Click on the icon on the right for the preprint pdf. This article documents the source of indeterminacy in Physics. My earlier article on Indeterminacy in Quantum Mechanics is being split into 4 separate titles. This is the first of those. It proves and describes the indeterminate existence of certain scalars used in Applied Mathematics and by extension in Theoretical Physics. This paper itself contains no physics. That will follow in the other titles.

StevieFaulkner@googlemail.com

The origin of indeterminacy within quantum physics and the mechanism of decision at measurement

The accompanying downloadable pdf is the full research paper resolving logical problems in Quantum Physics. Help yourself.

This study confirms that the conventionally formulated Quantum Mechanics with which we are familiar, is not the whole theory. The part that has been missing is not some undiscovered physics; it is Mathematical Logic. And in order to be entire, Physical Theory must embrace and incorporate it.

Amongst the discoveries of this research is foundation for The Quantum Logic. This is not an experimental logic or postulated ‘toy’; on the contrary, it is logic found present in the theory which Applied Mathematics cannot detect. It is found that Quantum Mechanics, conventionally prescribed by a collection of postulates, is a fragment of a larger theory, axiomatised under the Field Axioms. From first principles, these axioms derive equations of the subject, but show that different types of scalar: complex, real and rational, carry distinctly different logical validities. This is because the rational scalars exist as theorems of the Field Axioms, whereas existence of complex and real scalars can neither be proved nor disproved: they are mathematically undecidable.

In showing this, the role played by the square root of minus one is rigorously established. And in doing so, findings further establish that this research applies to all theories in Quantum Physics that employ scalar products between vectors in orthogonal spaces.

This new formalism resolves long standing logical anomalies for which the subject is notorious. It derives quantum indeterminacy, the existence of probability, Pythagorean addition of probability amplitudes and the mechanism of decision at measurement. It overturns two ‘intuitive fundamentalities’ and derives them instead: that observables are real values and entities must show particle behaviour.

The theory substantiates the 3-valued logic of Reichenbach, and via his work resolves ‘causal anomalies’ of complementarity and the EPR-paradox of action at a distance.

Theorems of Model Theory show that validities and indeterminacy of this axiomatised Quantum Mechanics go hand in hand with its theorems and undecidable sentences. These are interpreted as a causeology in Nature providing entities whose existence is caused and other entities whose existence is permitted. These mirror rational scalars, caused by the theory’s axiomatisation, and complex scalars permitted by the theory’s axiomatisation. The caused entities can be confirmed and witnessed; but the permitted entities can be neither confirmed nor denied.

What is especially interesting is, in contrast to the causative processes of classical physics where cause ascends through chains: …effect > caused-effect > caused-effect > caused-effect… ; there are quantum effects that are not caused by effects earlier in the chain but are permitted by them. In other words, there may be an effect in a chain for which no cause is traceable and for which information about the chain, upstream, is lost. To illustrate; in a wave/particle duality, all the various wavelengths that make up a wavepacket are caused effects; but probability amplitude is an uncaused permitted effect, unable to pass on the chain’s information to probability, which it in turn then causes. In addition, although all the momenta of a wavepacket are caused, and likewise so are the positions, the existence of both together is merely permitted.

Your feedback is welcome.  My email is at:

StevieFaulkner@googlemail.com

Who’s Afraid of a Big Black Hole?- Can Self Reference help?

The Horizon programme, “Who’s Afraid of a Big Black Hole?” was broadcast on 3rd November 2009. In a very clear and lucid account, the programme brings us up-to-date, on the state of science regarding the densest concentrations of mass in our Universe. After explaining the major advances, the programme focuses on difficulties in the theory relating to the singularity: that point at the centre of a black hole into which matter relentlessly cascades. Because it is an infinitesimal point, gravitation converges to infinity there; and as it stands, physical theory breaks down.

The converging gravitational field and the singularity was predicted by Einstein’s General Theory of Relativity, but because the singularity is a microscopic object, in its vicinity, Quantum Mechanics must also be satisfied. Nevertheless, for some decades, theorists have struggled without success to unify these two theories, and the programme concludes with those eminent scientists contributing to the programme, expressing their bemused but determined puzzlement as to where they should turn for inspiration on how to form one theory from the two.

While watching the programme, I knew I must make cosmologists aware of the idea that motivated my own studies, an idea of some merit and substance, because that idea has resulted in a thesis that derives the logic of Quantum Mechanics. It was this: questions along the lines: “What Physical Laws brought about The Beginning?” are misplaced on grounds of logic. No true beginning can have laws that precede it. Otherwise, what rules generated those laws… and so on, ad infinitum.

It was this thought, in 1997, that led me to consider the possibility of self-reference in physics. A friend then pointed me in the direction of Kurt Gödel’s theorems and they naturally led to a search for mathematical undecidability in Quantum Mechanics.

In the Horizon programme, parallels are drawn between the singularities of black holes and another singularity with similar conditions: the one from which the Big Bang exploded. Now of course, the singularity of the Big Bang may not have been The Beginning; but if it was we should not expect to find Physical Laws to have brought it about.

A logical alternative to this is that the Matter, the Spacetime and the Physical Laws all came into existence together in a self referent loop where each initiates the other: a loop that would otherwise have been hovering in some undecidable state.

Since 2006 I have had extremely good success in finding mathematical undecidability at the core of quantum mathematics. This has a logical form where products of orthogonal vectors are not axiomatic; yet the Field Axioms, the rules of addition and multiplication for scalar components of all mathematical objects, neither forbid nor accommodate these products.

Quantum Mechanics then develops into a theory that tells us where indeterminacy comes from, what happens to it at measurement, why the world we witness is real, the reason for particles, and why probability is not indeterminate. There are also signs that the indeterminacy of chaos follows from a related but quite separate undecidability. Most remarkably, once seen in terms of its logical form, Quantum Mechanics becomes an elegantly intuitive theory.

Mathematical undecidability lies within systems whose operation has no option but to follow a deficient set of rules. My suggestion is that at the singularity there is such deficiency and that this is a good candidate worthy of investigation.

Causeology in Quantum Physics

My work finds that hidden within Nature there is a causeology where there are entities whose existence is CAUSED and other entitiies whose existence is PERMITTED.  The caused entities can be confirmed and witnessed; but the permitted entities can be neither confirmed nor denied.

What is interesting is that unlike the causative processes with which we are accustommed in everyday experience, cause acends through chains:   cause, effect…cause, effect…cause,effect…; these quantum causes derive and materialise, not from caused entities up the chain, but permitted entities that are not caused. To illustrate, in a wave/particle duality, all the various wavelengths that make up a wavepacket are caused entities; but the interfering sums of these waves are permitted.  From these permitted interfernces derive caused probabilities.  The interfernces constitute a break in the causal chain. This claim is born out in the mathematics using Model Theory.

In the mathematics, this regime has its origin in the way numbers add and multiply.  The rules are the Field Axioms. Some numbers are caused by these axioms: the rational numbers exist as theorems; others are permitted. Juxtaposed against this system, are  products between orthogonal vectors which force the existence of certain permitted numbers. These orthogonal vectors are objects that, when multiplied with themselves give one, and, when multiplied with each other give zero.

The results lead to a 3-valued logic of true, false and indeterminate.  This was anticipated by Hans Reichenbach.  His 1944 book details the effects of this logic.  It does indeed resolve the paradoxes in quantum physics.  What Reichenbach had no idea of is where his logic came from…so it has been generally ignored.

My work finds that hidden within Nature there is a causeology where there are entities whose existence is caused and other entities whose existence is permitted.  The caused entities can be confirmed and witnessed; but the permitted entities can be neither confirmed nor denied.

What is interesting is that in contrast to the causative processes in classical physics where cause ascends through chains:   cause, effect…cause, effect…cause, effect…; there are quantum entities that derive, not from caused entities up the chain, but permitted entities that are not caused. To illustrate; in a wave/particle duality, all the various wavelengths that make up a wavepacket are caused entities; but the interfering sums of these waves are permitted.  From these permitted interferences derive caused probabilities.  So the interferences constitute a break in the causal chain. This claim is born out in the mathematics using Model Theory.

In the mathematics, this causal regime has its origin in the way numbers add and multiply.  The rules for this adding and multiplying are the Field Axioms. Some numbers are caused by these axioms: the rational numbers exist as theorems; other numbers with imaginary components, are permitted. Juxtaposed against this axiomatic system, are products between orthogonal vectors which force the existence of imaginary numbers which are merely permitted. The orthogonal vectors are objects that, when multiplied with themselves give one, and, when multiplied with each other give zero

The full causeology is 3-valued: cause, deny, permit.  These go hand in hand with a 3-valued logic of true, false and indeterminate.   This 3-valued logic was anticipated by Hans Reichenbach.  His 1944 book details the effects of this logic.  It does indeed resolve the paradoxes in quantum physics.  What Reichenbach had no idea of was where his logic came from…so it has been generally put to one side; now it has foundation.

Indeterminacy produced by Quantum Mechanics embedded in Mathematical Logic